Solving second order PDE

Question

I have the PDE $x^2U_{xx}-2xyU_{xy}+y^2U_{yy}+xU_x + yU_y = 0.$

I thought this could be handled by a change of variables $X = ln|x|$ , $Y=ln|y|$ , but this approach gets me stuck pretty fast.

How can I solve it?

Answer 1

The change of variables $\begin{cases} X=\ln|x|\\ Y=\ln|y| \end{cases}$ transforms the PDE into : $$U_{XX}-2U_{XY}+U_{YY}=0$$ A second change of variables : $\begin{cases} s=X+Y\\ t=X-Y \end{cases}$ reduces the PDE to : $$U_{tt}=0$$ The double integration of this very simple differential equation gives : $$U(s,t)=f(s)t+g(s)$$ where $f$ and $g$ are any derivable functions. Then, coming back to the preceeding variables : $$U(X,Y)=(X-Y)f(X+Y)+g(X+Y)$$ $$U(x,y)=\ln|\frac{x}{y}| f(\ln|xy|)+g(\ln|xy|)$$ Or, on an equivalent form with any derivable functions $F$ and $G$ : $$U(x,y)=\ln\left|\frac{x}{y}\right|\: F(xy)+G(xy)$$